Beyond tail median and conditional tail expectation: extreme risk estimation using tail Lp optimisation

Transcription

1 Beyond tail median and conditional tail expectation: extreme ris estimation using tail Lp optimisation Laurent Gardes, Stephane Girard, Gilles Stupfler To cite this version: Laurent Gardes, Stephane Girard, Gilles Stupfler. Beyond tail median and conditional tail expectation: extreme ris estimation using tail Lp optimisation <hal v2> HAL Id: hal Submitted on 28 Jan 209 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Beyond tail median and conditional tail expectation: extreme ris estimation using tail L p optimisation Laurent Gardes, Stéphane Girard 2 & Gilles Stupfler 3 Université de Strasbourg & CNRS, IRMA, UMR 750, 7 rue René Descartes, Strasbourg Cedex, France 2 Université Grenoble Alpes, INRIA, CNRS, Grenoble INP, LJK, Grenoble, France 3 School of Mathematical Sciences, University of Nottingham, University Par Campus, Nottingham NG7 2RD, United Kingdom Abstract. The Conditional Tail Expectation is an indicator of tail behaviour that, contrary to the quantile or Value-at-Ris, taes into account the frequency of a tail event together with the probabilistic behaviour of the variable of interest on this event. However, the asymptotic normality of the empirical Conditional Tail Expectation estimator requires that the underlying distribution possess a finite variance; this can be a strong restriction in the heavy-tailed models of interest in insurance and finance. One solution when this assumption fails could be to use the more robust Median Shortfall, but being a quantile, this quantity only gives information about the frequency of a tail event. We construct a synthetic class of tail L p medians, encompassing the Median Shortfall and Conditional Tail Expectation. We show that, for < p < 2, a tail L p median depends on both the frequency and magnitude of tail events, and its empirical estimator is, within the range of the data, asymptotically normal under a condition weaer than a finite variance. We extrapolate this estimator and another technique to extreme levels using the heavy-tailed framewor. The estimators are showcased on a simulation study and on a set of real fire insurance data showing evidence of a very heavy right tail. Keywords: asymptotic normality, Conditional Tail Expectation, extreme value statistics, heavy-tailed distribution, L p optimisation, Median Shortfall, semiparametric extrapolation.

3 Introduction A precise assessment of extreme ris is a crucial question in a number of fields of statistical applications. In actuarial science and finance, especially, a major question is to get a good understanding of potential extreme claims and losses that would constitute a threat to the survival of a company. This has historically been done by simply using a quantile, also called Value-at-Ris VaR in the actuarial and financial world. The quantile of a real-valued random variable X at level α 0, is, by definition, the lowest value qα which is exceeded by X with probability not larger than α. As a consequence, quantiles only provide an information about the frequency of a tail event; in particular, the quantile qα gives no indication about how heavy the right tail of X is or, perhaps more concretely, as to what a typical value of X above qα would be. From a ris assessment perspective, this is clearly an issue, as quantiles allow to quantify what a risy situation is i.e. the quantile level, but not what the consequences of a risy situation would be i.e. the behaviour of X beyond the quantile level. The notion of Conditional Tail Expectation CTE precisely addresses this point. This ris measure, defined as CTEα := E[X X > qα], is exactly the average value of X given that X > qα. When X is continuous, CTEα coincides with the so-called Expected Shortfall ESα, also nown as Tail Value-at-Ris or Conditional Value-at-Ris and discussed in [2, 44, 45], which is the average value of the quantile function τ qτ over the interval [α,. The potential of CTE for use in actuarial and financial ris management has been considered by a number of studies, such as [0, 23, 46, 48]. Outside academic contexts, the CTE ris measure is used in capital requirement calculations by the Canadian financial and actuarial sectors [33], as well as for guaranteeing the sustainability of life insurance annuities in the USA [42]. European regulators, via the Basel Committee on Baning Supervision, also recently recommended to use CTE rather than VaR in internal maret ris models, see [5]. Whether the CTE or VaR should be used in a given financial application is still very much up for debate: we point to, among others, [4, 6, 22, 26, 39] who discuss the various intrinsic axiomatic properties of CTE and/or VaR and their practical interpretations. Aside from axiomatic considerations, the estimation of the CTE by the empirical conditional tail moment requires a finite tail second moment if this estimator is to be asymptotically normal. This should of course be expected, since a condition for the sample average to be an asymptotically 2

4 normal estimator of the sample mean is precisely the finiteness of the variance. In heavy-tailed models, which are of interest in insurance/finance and that will be the focus of this paper, such moment restrictions can be essentially reformulated in terms of a condition on the tail index γ of X. The variable X is heavy-tailed with index γ > 0 if its distribution behaves approximately lie a power law distribution with exponent /γ: PX > x = x /γ lx for x large enough, where l is a slowly varying function at infinity, namely ltx/lt as t for any x > 0; see [6, p.57]. In such a model, the finite second moment condition is violated when the tail index γ, controlling the right tail heaviness of X, is such that γ > /2. At the finite-sample level, it is often desirable to have a finite fourth moment; this stronger condition is violated whenever γ > /4. Table in the simulation study of [9] shows a quite strong deterioration in the finitesample performance of the empirical conditional tail moment as γ increases to /4. A number of actuarial and financial data sets have been found to violate such integrability assumptions: we refer to, for instance, the Danish fire insurance data set considered in Example 4.2 of [43], emerging maret stoc returns data [30, 40] and exchange rates data [3]. The deterioration in finite-sample performance as the number of finite moments decreases is due in no small part to increased variability in the right tail of X as γ increases, see Section 3. in [20]. To counteract this variability, one could move away from the CTE and use the Median Shortfall MS see [38, 39], just as one can use the median instead of the mean for added robustness. However, MSα is nothing but q[ + α]/2 and as such, similarly to VaR, does not contain any information on the behaviour of X beyond its value. Our goal here is to design a class of tail indicators that realise a compromise between the sensitivity of the Conditional Tail Expectation and the robustness of the Median Shortfall. We will start by showing that these two quantities can be obtained in a new unified framewor, which we call the class of tail L p medians. A tail L p median at level α 0, is obtained, for p, by minimising an L p moment criterion which only considers the probabilistic behaviour of X above a quantile qα. As such, it satisfies a number of interesting properties we will investigate. Most importantly, the class of tail L p medians is lined to the notion of L p quantiles introduced by [3] and recently studied in heavy-tailed framewors by [5], a tail L p median being the L p quantile of order /2 or L p median of the variable X X > qα. In particular, the classical median of X is the L median and the mean of X is the L 2 median or expectile of order /2, in the 3

5 terminology of [4], and we will show that similarly, the Median Shortfall is the tail L median, while the Conditional Tail Expectation is the tail L 2 median. Lie L p quantiles, tail L p medians with p > depend on both the frequency of the event {X > qα} and the actual behaviour of X beyond qα. At the technical level, a condition for a tail L p median to exist is that γ < /p, and it can be empirically estimated at high levels by an asymptotically Gaussian estimator if γ < /[2p ]. When p, 2, the tail L p median and the L p quantile therefore simultaneously exist and are accurately estimable in a wider class of models than the CTE is. However, there are a number of differences between tail L p medians and L p quantiles. For example, the theoretical analysis of the asymptotic behaviour of tail L p medians, as α where throughout the paper, denotes taing a left limit at, is technically more complex than that of L p quantiles. The asymptotic results that arise show that the tail L p median at level α is asymptotically proportional to the quantile qα, as α, through a non-explicit but very accurately approximable constant, and the remainder term in the asymptotic relationship is exclusively controlled by extreme value parameters of X. This stands in contrast with L p quantiles, which also are asymptotically proportional to the quantile qα through a simpler constant, although the remainder term crucially features the expectation and left-tail behaviour of X. The remainder term plays an important role in the estimation, as it determines the bias term in our eventual estimators, and we will then argue that the extreme value behaviour of tail L p medians is easier to understand and more natural than that of L p quantiles. We will also explain why, for heavy-tailed models, extreme tail L p medians are able to interpolate monotonically between extreme MS and extreme CTE, as p varies in, 2 and for γ <. By contrast, L p quantiles are nown not to interpolate monotonically between quantiles and expectiles see Figure in [5]. The interpolation property also maes it possible to interpret an extreme tail L p median as a weighted average of extreme MS and extreme CTE. This is liely to be helpful as far as the practical applicability of tail L p medians is concerned, to the extent that it allows for a simple choice of p reflecting a pre-specified compromise between the extreme MS and CTE. We shall then examine how to estimate an extreme tail L p median. This amounts to estimating a tail L p median at a level α = α n as the size n of the sample of data tends to infinity. We start by suggesting two estimation methods in the so-called intermediate case n α n. Although the design stage of our tail L p median estimators has similarities with that of the L p quantile estimators in [5], the investigation of the asymptotic properties of the tail L p median estimators 4

6 is more challenging and technically involved. These methods will provide us with basic estimators that we will extrapolate at a proper extreme level α n, which satisfies n α n c [0,, using the heavy-tailed assumption. We will, as a theoretical byproduct, demonstrate how our results mae it possible to recover or improve upon nown results in the literature on Conditional Tail Expectation. Our final step will be to assess the finite-sample behaviour of the suggested estimators. Our focus will not be to consider heavy-tailed models with a small value of γ: for these models and more generally in models with finite fourth moment, it is unliely that any improvement will be brought on the CTE, whether in terms of quality of estimation or interpretability. Our view is rather that the use of tail L p medians with p, 2 will be beneficial for very heavy-tailed models, in which γ is higher than the finite fourth moment threshold γ = /4, and possibly higher than the finite variance threshold γ = /2. For such values of γ and with an extreme level set to be α n = /n a typical choice in applications, see e.g. recently [3,, 2, 27], we shall then evaluate the finite-sample performance of our estimators on simulated data sets, as well as on a real set of fire insurance data featuring an estimated value of γ larger than /2. The paper is organised as follows. We first give a rigorous definition of our concept of tail L p median and state some of its elementary properties in Section 2. Section 3 then focuses on the analysis of asymptotic properties of the population tail L p median, as α, in heavy-tailed models. Estimators of an extreme tail L p median, obtained by first constructing two distinct estimators at intermediate levels which are then extrapolated to extreme levels, will be discussed in Section 4. A simulation study of the finite-sample performance of our estimators is presented in Section 5, and an application to real fire insurance data is discussed in Section 6. Proofs and auxiliary results are deferred to the Appendix. 2 Definition and first properties Let X be a real-valued random variable with distribution function F and quantile function q given by qα := inf{x R F x α}. It is assumed throughout the paper that F is continuous, so that F qα = α for any α 0,. Our construction is motivated by the following two observations. Firstly, the median and mean of X can respectively be obtained by 5

7 minimising an expected absolute deviation and expected squared deviation: q/2 = arg min E X m X and EX = m R arg min E X m 2 X 2 provided E X <. m R The first identity is shown in e.g. [36, 37]; the minimiser on the right-hand side thereof may actually not be unique although it is if F is strictly increasing. Our convention throughout this paper will be that in such a situation, the minimiser is taen as the smallest possible minimiser, maing the identity valid in any case. Note also that subtracting X resp. X 2 within the expectation in the cost function for q/2 resp. EX maes it possible to define this cost function, and therefore its minimiser, without assuming any integrability condition on X resp. by assuming only E X <, as a consequence of the triangle inequality resp. the identity X m 2 X 2 = mm 2X. These optimisation problems extend their arguably better-nown formulations q/2 = arg min m R E X m and EX = arg min EX m 2 m R which are only well-defined when E X < and EX 2 < respectively. Secondly, since F is continuous, the Median Shortfall MSα = q[ + α]/2 is the median of X given X > qα, see Example 3 in [39]. Since CTEα is the expectation of X given X > qα, we find that MSα = arg min E X m X X > qα and CTEα = m R arg min E X m 2 X 2 X > qα provided E X X > qα <. m R Our construction now encompasses these two quantities by replacing the absolute or squared deviations by power deviations. Definition. The tail L p median of X, of order α 0,, is when it exists m p α = arg min E X m p X p X > qα. m R Let us highlight the following important connection between the tail L p median and the notion of L p quantiles: recall, from [3, 5], that an L p quantile of order τ 0, of a univariate random variable Y, with E Y p <, is defined as q τ p = arg min E τ {Y q} Y q p τ {Y 0} Y p. q R 6

8 Consequently, the L p median of Y, obtained for τ = /2, is q /2 p = arg min E Y q p Y p. q R For an arbitrary p, the tail L p median m p α of X is then exactly an L p median of X given that X > qα. This construction of m p α as an L p median given the tail event {X > qα} is what motivated the name tail L p median for m p α. We underline again that subtracting X p inside the expectation in Definition above maes the cost function well-defined whenever E X p X > qα is finite or equivalently EX p + <, where X + := maxx, 0. This is a straightforward consequence of the triangle inequality when p = ; when p >, this is a consequence of the fact that the function x x p is continuously differentiable with derivative x p x p signx, together with the mean value theorem. If X is moreover assumed to satisfy EX p + <, then the definition of m p α is equivalently and perhaps more intuitively m p α = arg min E X m p X > qα. m R The following result shows in particular that if EX p + < for some p, then the tail L p median always exists and is characterised by a simple equation. Especially, for p, 2, the tail L p median m p α exists and is unique under a weaer integrability condition than the assumption of a finite first tail moment which is necessary for the existence of CTEα. Proposition. Let p. Pic α 0, and assume that EX p + <. Then: i The tail L p median m p α exists and is such that m p α > qα. ii The tail L p median is equivariant with respect to increasing affine transformations: if m X p α is the tail L p median of X and Y = ax + b with a > 0 and b R, then the tail L p median m Y p α of Y is m Y p α = a m X p α + b. iii When p >, the tail L p median m p α is the unique m R solution of the equation E[m X p {qα<x<m} ] = E[X m p {X>m} ]. iv The tail L p median defines a monotonic functional with respect to first-order stochastic dominance: if Y is another random variable such that EY p + < then t R, PX > t PY > t m X p α m Y p α. 7

9 v When E X p <, the function α m p α is nondecreasing on 0,. Let us highlight that, in addition to these properties, we have most importantly MSα = m α and CTEα = m 2 α. 2 In other words, the Median Shortfall is the tail L median and the Conditional Tail Expectation is the tail L 2 median: the class of tail L p medians encompasses the notions of MS and CTE. We conclude this section by noting that, lie MS, the tail L p median is not subadditive for < p < 2. Our objective here is not, however, to construct an alternative class of ris measures which have perfect axiomatic properties. There have been several instances when non-subadditive measures were found to be of practical value: besides the non-subadditive VaR and MS, L p quantiles define, for < p < 2, non-subadditive ris measures that can be fruitfully used for, among others, the bactesting of extreme quantile estimates see [5]. This wor is, rather, primarily intended to provide an interpretable middleway between the ris measures MSα and CTEα, for a level α close to. This will be useful when the number of finite moments of X is low typically, less than 4 because the estimation of CTEα, for high α, is then a difficult tas in practice. 3 Asymptotic properties of an extreme tail L p median It has been found in the literature that heavy-tailed distributions generally constitute appropriate models for the extreme value behaviour of actuarial and financial data, and particularly for extremely high insurance claims and atypical financial log-returns see, e.g., [2, p.9] and [43, p.]. Our first focus is therefore to analyse whether m p α, for p, 2, does indeed provide a middle ground between MSα and CTEα, as α, in heavy-tailed models. This is done by introducing the following regular variation assumption on the survival function F := F of X. C γ There exists γ > 0 such that for all x > 0, lim t F tx/f t = x /γ. In other words, condition C γ means that the function F is regularly varying with index /γ in a neighbourhood of +, see [8]; for this reason, condition C γ is equivalent to assuming. Note that, most importantly for our purposes, assuming this condition is equivalent to supposing that the tail quantile function t Ut := q t of X is regularly varying with tail index γ. We also remar that when C γ is satisfied then X does not have any finite conditional tail moments of order larger than /γ; similarly, all conditional tail moments of X of order smaller 8

10 than /γ are finite a rigorous statement is Exercise.6 in [28]. Since the existence of a tail L p median requires finite conditional tail moments of order p, this means that our minimal woring condition on the pair p, γ should be γ < /p, a condition that will indeed appear in many of our asymptotic results. We now provide some insight into what asymptotic result on m p α we should aim for under condition C γ. A consequence of this condition is that above a high threshold u, the variable X/u is approximately Pareto distributed with tail index γ, or, in other words: X x >, P qα > x X qα > x /γ as α. This is exactly the first statement of Theorem.2. in [28]. The above conditional Pareto approximation then suggests that when α is close enough to, the optimisation criterion in Definition can be approximately written as follows: m p α qα arg min E [ Z γ M p Z γ p ] M R where Z γ has a Pareto distribution with tail index γ. For the variable Z γ, we can differentiate the cost function and use change-of-variables formulae to get that the minimiser M of the righthand side should satisfy the equation g p,γ /M = Bp, γ p +, where g p,γ t := t u p u /γ du for t 0, and Bx, y = 0 vx v y dv denotes the Beta function. In other words, and for a Pareto random variable, we have m p α qα = κp, γ where κp, γ := g p,γbp, γ p + and g p,γ denotes the inverse of the decreasing function g p,γ on 0,. Our first asymptotic result states that this proportionality relationship is still valid asymptotically under condition C γ. Proposition 2. Suppose that p and C γ holds with γ < /p. Then: m p α lim α qα = κp, γ. It follows from Proposition 2 that a tail L p median above a high exceedance level is approximately a multiple of this exceedance level. This first-order result is similar in spirit to other asymptotic proportionality relationships lining extreme ris measures to extreme quantiles: we refer to [4] for a result on extreme expectiles and [5] on the general class of L p quantiles, as well as to [9, 49, 50] for a similar analysis of extreme Wang distortion ris measures. A consequence of this result is 9

11 that, similarly to extreme L p quantiles, an extreme tail L p median contains both the information contained in the quantile qα plus the information on tail heaviness provided by the tail index γ. This point shall be further used and discussed in Sections 4.2 and 5. Let us also highlight that Proposition 2 does not hold true for γ = /p, since κp, γ is then not well-defined. The asymptotic proportionality constant κp, γ 0, does not have a simple closed form in general, due to the complicated expression of the function g p,γ. It does however have a nice explicit expression in the two particular cases p = and p = 2. For p =, we have κ, γ = 2 γ, see Lemma 2ii in Appendix A.2. This clearly yields the same equivalent as the one obtained using the regular variation of the tail quantile function U with index γ, i.e. in virtue of 2: m α qα = q + α/2 qα = U2 α U α 2γ as α. When p = 2 and γ 0,, κ2, γ = γ, see Lemma 3ii in Appendix A.2. Since in this case, m 2 α is nothing but CTEα by 2, Proposition 2 agrees here with the asymptotic equivalent of CTEα in terms of the exceedance level qα, see e.g. [32]. For other values of p, an accurate numerical computation of the constant κp, γ can be carried out instead. Results of such numerical computations on the domain p, γ [, 2] 0, are included in Figure. One can observe from this Figure that the functions p κp, γ and γ κp, γ seem to be both decreasing. This and 2 entail in particular that, for all p, p 2, 2 such that p < p 2, we have, for α close enough to : MSα = m α < m p α < m p2 α < m 2 α = CTEα. The tail L p median m p α can therefore be seen as a ris measure interpolating monotonically between MSα and CTEα, at a high enough level α. Actually, Proposition 2 yields, for all p [, 2] and γ < /p that, as α, m p α λp, γmsα + [ λp, γ]cteα, 3 where the weighting constant λp, γ [0, ] is defined by m p α CTEα λp, γ := lim α MSα CTEα γ/κp, γ =. 4 2 γ γ Extreme tail L p medians of heavy-tailed models can then be interpreted, for p, 2, as weighted averages of extreme Median Shortfall and extreme Conditional Expectation at the same level. It 0

12 should be noted that, by contrast, the monotonic interpolation property and hence the weighted average interpretation is demonstrably false in general for L p quantiles, as is most easily seen from Figure in [5]: this Figure suggests that high L p quantiles define, for γ close to /2, a decreasing function of p when it is close to and an increasing function of p when it is close to 2. The fact that γ κp, γ is decreasing, meanwhile, can be proven rigorously by noting that its partial derivative κ/ γ is negative see Theorem 2 below. More intuitively, the monotonicity of γ κp, γ can be seen as a consequence of the heavy-tailedness of the distribution function F. Indeed, we saw that heuristically, as α, m p α qα arg min E [ Z γ M p Z γ p ] M R where Z γ is a Pareto random variable with tail index γ. When γ increases, the random variable Z γ tends to return higher values because its survival function PZ γ > z = z /γ for z > is an increasing function of γ. We can therefore expect that, as γ increases, a higher value of M = m p α/qα will be needed in order to minimise the above cost function..0 3/ gamma /2 0.4 / p Figure : Behaviour of p, γ κp, γ for p [, 2] and γ 0,.

13 Our next goal is to derive an asymptotic expansion of the tail L p median m p α, relatively to the high exceedance level qα. This will be the ey theoretical tool maing it possible to analyse the asymptotic properties of estimators of an extreme tail L p median. For this, we need to quantify precisely the error term in the convergence given by Proposition 2, and this prompts us to introduce the following second-order regular variation condition: C 2 γ, ρ, A The function F is second-order regularly varying in a neighbourhood of + with index /γ < 0, second-order parameter ρ 0 and an auxiliary function A having constant sign and converging to 0 at infinity, that is, [ ] F tx x > 0, lim t A/F t F t x /γ where the right-hand side should be read as x /γ log x γ 2 when ρ = 0. = x /γ xρ/γ, γρ This standard condition on F controls the rate of convergence in C γ: the larger ρ is, the faster the function A converges to 0 since A is regularly varying with index ρ, in virtue of Theorems and in [28] and the smaller the error in the approximation of the right tail of X by a purely Pareto tail will be. Further interpretation of this assumption can be found in [6] and [28] along with numerous examples of commonly used continuous distributions satisfying it. Let us finally mention that it is a consequence of Theorem in [28] that C 2 γ, ρ, A is actually equivalent to the following, perhaps more usual extremal assumption on the tail quantile function U: [ ] Utx x > 0, lim t At Ut xγ = x γ xρ. 5 ρ In 5, the right-hand side should be read as x γ log x when ρ = 0 similarly, throughout this paper, quantities depending on ρ are extended by continuity using their left limit as ρ 0. The next result is the desired refinement of Proposition 2 giving the error term in the asymptotic proportionality relationship lining m p α to qα when α. Proposition 3. Suppose that p and C 2 γ, ρ, A holds with γ < /p. Then, as α, m p α qα = + [Rp, γ, ρ + o]a α κp, γ with Rp, γ, ρ = [κp, γ] ρ /γ κp, γ ρ [ Bp, ργ p + g p p,γ/ ρ κp, γ ]. γρ 2

14 This result is similar in spirit to second-order results that have been shown for other extreme ris measures: we refer again to [49, 50], as well as to [4, 5, 9] for analogue results used as a basis to carry out extreme-value based inference on other types of indicators. It should, however, be underlined that the asymptotic expansion of an extreme tail L p median depends solely on the extreme parameters γ, ρ and A, along with the power p. By contrast, the asymptotic expansion of an extreme L p quantile depends on the expectation and left-tail behaviour of X, which are typically considered to be irrelevant to the understanding of the right tail of X. From an extreme value point of view, the asymptotic expansion of extreme tail L p medians is therefore easier to understand than that of L p quantiles. Statistically speaing, it also implies that there are less sources of potential bias in extreme tail L p median estimation than in extreme L p quantile estimation. Both of these statements can be explained by the fact that the tail L p median is constructed exclusively on the event {X > qα}, while the equation defining an L p quantile q p α is αeq p α X p {X<qpα} = αex q p α p {X>qpα} see [3, Section 2]. The left-hand side term ensures that the central and left-tail behaviour of X will necessarily have an influence on the value of any L p quantile, even at an extreme level. The construction of a tail L p median as an L p median in the right tail of X removes this issue. Lie the asymptotic proportionality constant κp, γ on which it depends, the remainder term Rp, γ, ρ does not have an explicit form in general. That being said, R, γ, ρ and R2, γ, ρ have simple explicit values: for p =, R, γ, ρ = 2 ρ /ρ with the convention x ρ /ρ = log x for ρ = 0, see Lemma 2iii in Appendix A.2. We then find bac the result that comes as an immediate consequence of 2 and our second-order condition C 2 γ, ρ, A, via 5: m α = U2 α qα U α = 2γ + A α 2ρ + o. ρ For p = 2 and γ 0,, 2 and 5 suggest that: m 2 α qα = U α x dx U α x = 2 γ + γ ρ A α + o which coincides with Proposition 3, given that R2, γ, ρ = / γ ρ from Lemma 3iii in Appendix A.2. We close this section by noting that all our results, and indeed the practical use of the tail L p median more generally, depend on the fixed value of the constant p. 3 Just as when using

15 L p quantiles, the choice of p in practice is a difficult but important question. Although [3] introduced L p quantiles in the context of testing for symmetry in non-parametric regression, it did not investigate the question of the choice of p. In [5], extreme L p quantiles were used as vehicles for the estimation of extreme quantiles and expectiles and for extreme quantile forecast validation; in connection with the latter, it is observed that neither p = nor p = 2 provide the best performance in terms of forecast, but no definitive conclusion is reached as to which value of p should be chosen see Section 7 therein. For extreme tail L p medians, which unlie extreme L p quantiles satisfy an interpolation property, we may suggest a potentially simpler and intuitive way to choose p. Recall the weighted average relationship 3: m p α λp, γmsα + [ λp, γ]cteα, with λp, γ = γ/κp, γ 2 γ γ for α close to. In practice, given the estimated value of γ, and a pre-specified weighting constant λ 0 indicating a compromise between robustness of MS and sensitivity of CTE, one can choose p = p 0 as the unique root of the equation λp, γ = λ 0 with unnown p. Although λp, γ does not have a simple closed form, our experience shows that this equation can be solved very quicly and accurately with standard numerical solvers. This results in a tail L p median m p0 α satisfying, for α close to, m p0 α λ 0 MSα + λ 0 CTEα. The interpretation of m p0 α is easier than that of a generic tail L p median m p α due to its explicit and fully-determined connection with the two well-understood quantities MSα and CTEα. The question of which weighting constant λ 0 should be chosen is itself difficult and depends on the requirements of the situation at hand; our real data application in Section 6 provides an illustration with λ 0 = /2, corresponding to a simple average between extreme MS and CTE. 4 Estimation of an extreme tail L p median Suppose that we observe a random sample X,..., X n of independent copies of X, and denote by X,n X n,n the corresponding set of order statistics arranged in increasing order. Our goal in this section is to estimate an extreme tail L p median m p α n, where α n as n. The final aim is to allow α n to approach at any rate, covering the cases of an intermediate tail 4

16 L p median with n α n and of a proper extreme tail L p median with n α n c, where c is some finite positive constant. 4. Intermediate case: direct estimation by empirical L p optimisation Recall that the tail L p median m p α n is, by Definition, m p α n = arg min E X m p X p X > qα n. m R Assume here that n α n, so that n α n > 0 eventually. We can therefore define a direct empirical tail L p median estimator of m p α n by minimising the above empirical cost function: m p α n = arg min m R n α n n α n We now pave the way for a theoretical study of this estimator. i= X n i+,n m p X n i+,n p. 6 The ey point is that since normalising constants and shifts are irrelevant in the definition of the empirical criterion, we clearly have the equivalent definition Consequently with ψ n u; p = m p α n = arg min m R p[m p α n ] p n α n p[m p α n ] p i= mp α n n αn m p α n n α n i= X n i+,n m p X n i+,n m p α n p. = arg min ψ n u; p u R X n i+,n m p α n um p pα n X n i+,n m p α n p. n αn 7 Note that the empirical criterion ψ n u; p is a continuous and convex function of u, so that the asymptotic properties of the minimiser follow directly from those of the criterion itself by the convexity lemma of [25] see also [35]. The empirical criterion, then, is analysed by using its continuous differentiability for p > in order to formulate an L p analogue of Knight s identity [34] and divide the wor between, on the one hand, the study of a n α n consistent and asymptotically Gaussian term which is an affine function of u and, on the other hand, a bias term which converges to a nonrandom multiple of u 2. Further technical details are provided in the Appendix, and especially Lemmas 6, 7, 9 and. 5

17 This programme of wor is broadly similar to that of [5] for the convergence of the direct intermediate L p quantile estimator. The difficulty in this particular case, however, is twofold: first, the affine function of u is a generalised L statistic in the sense of e.g. [9] whose analysis requires delicate arguments relying on the asymptotic behaviour of the tail quantile process via Theorem 5..4 p.6 in [28]. For L p quantiles, this is not necessary because the affine term is actually a sum of independent, identically distributed and centred variables. Second, the bias term is essentially a doubly integrated oscillation of a power function with generally noninteger exponent. The examination of its convergence requires certain precise real analysis arguments which do not follow from those developed in [5] for the asymptotic analysis of intermediate L p quantiles. With this in mind, the asymptotic normality result for the direct intermediate tail L p median estimator m p α n is the following. Theorem. Suppose that p and C 2 γ, ρ, A holds with γ < /[2p ]. Assume further that α n is such that n α n and n α n A α n = O. Then we have, as n : Here V p, γ = V p, γ/v 2 p, γ with mp α n n αn m p α n d N 0, V p, γ. V p, γ = [κp, γ]/γ γ B2p, γ 2p + g 2p,γ κp, γ + [ κp, γ] 2p while V 2 p, γ is defined by: V 2, γ = /γ 2 and, for p >, V 2 p, γ = p γ [κp, [ γ]/γ Bp, γ p g p,γ κp, γ ] 2. Moreover, the functions V 2, γ and V, γ defined this way are right-continuous at. The asymptotic variance V p, γ has a rather involved expression. Figures 2 and 3 provide graphical representations of this variance term. It can be seen on Figure 2 that the function γ V p, γ appears to be increasing. This reflects the increasing tendency of the underlying distribution to generate extremely high observations when the tail index increases see, among others, Section 3. in [20], thus increasing the variability of the empirical criterion ψ n ; p and consequently that of its minimiser. It is not, however, clear from this figure that the function p V p, γ is monotonic, lie the proportionality constant κ was. It turns out that, somewhat surprisingly, the function p V p, γ is not in general a monotonic 6

18 gamma p Figure 2: Behaviour of the asymptotic variance V of the direct empirical tail L p median estimator. Top panel: a plot of the function γ V p, γ, on the interval 0, /2, for p {,.2,.4,.6,.8, 2}. Bottom panel: a plot of the function p, γ V p, γ on the domain [,.8] [0.25, 0.5]. 7

19 Vp, p Figure 3: Behaviour of the function p V p, 0.3 on the interval [,.8]. function of p, and an illustration is provided for γ = 0.3 in Figure 3. A numerical study, which is not reported here, actually shows that we have V p, γ < V, γ for any p, γ,.2] [0.25, 0.5]. This suggests that for all heavy-tailed distributions having only a second moment an already difficult case as far as estimation in heavy-tailed models is concerned, a direct L p tail median estimator with p,.2] will have a smaller asymptotic variance than the empirical L tail median estimator, or, in other words, the empirical Median Shortfall. We conclude this section by noting that, lie the constants appearing in our previous asymptotic results, the variance term V p, γ has a simple expression when p = or p = 2. In the case p =, we have V, γ = 2γ 2 by Lemma 2iv in Appendix A.2. Statement 2 suggests that this should be identical to the asymptotic variance of the high quantile estimator MSα n = X n n αn/2,n, and indeed MSαn n αn MSα n = 2 n αn 2 Xn n αn/2,n q α n /2 d N 0, 2γ 2 by Theorem p.52 in [28]. For p = 2, we have V 2, γ = 2γ 2 γ/ 2γ by Lemma 3iv in Appendix A.2. By 2, we should expect this constant to coincide with the asymptotic variance 8

20 of the empirical counterpart of the Conditional Tail Expectation, namely ĈTEα n = n α n X n i+,n. n α n This is indeed the case as Corollary in [7] shows; see also Theorem 2 in [8]. In particular, the function γ V 2, γ tends to infinity as γ /2, reflecting the increasing difficulty of estimating a high Conditional Tail Expectation by its direct empirical counterpart as the right tail of X gets heavier. i= 4.2 Intermediate case: indirect quantile-based estimation We can also design an estimator of m p α n based on the asymptotic equivalence between m p α n and qα n that is provided by Proposition 2. Indeed, since this result suggests that m p α/qα /κp, γ when α with denoting asymptotic equivalence throughout, it maes sense to build a plug-in estimator of m p α n by setting m p α n = qα n /κp, γ n, where qα n and γ n are respectively two consistent estimators of the high quantile qα n and of the tail index γ. Since we wor here in the intermediate case n α n, we now that the sample counterpart X nαn,n of qα n is a relatively consistent estimator of qα n, see Theorem 2.4. in [28]. This suggests to use the estimator m p α n := X nα n,n κp, γ n. 8 Our next result analyses the asymptotic distribution of this estimator, assuming that the pair γ n, X nαn,n is jointly n α n consistent. Theorem 2. Suppose that p and C 2 γ, ρ, A holds with γ < /p. Assume further that α n is such that n α n and n α n A α n λ R, and that n αn γ n γ, X nα n,n d ξ, ξ 2 qα n where ξ, ξ 2 is a pair of nondegenerate random variables. Then we have, as n : mp α n n αn m p α n d σp, γξ + ξ 2 λrp, γ, ρ, with the positive constant σp, γ being σp, γ = κ p, γ. In other words, κp, γ γ σp, γ = Bp, γ p + [Ψγ + Ψγ p + ] κp,γ up u /γ logu du γ 2 [ κp, γ] p [κp, γ] /γ where Ψx = Γ x/γx is Euler s digamma function. 9

21 Again, the constant σp, γ does not generally have a simple explicit form, but we can compute it when p = or p = 2. Lemma 2v shows that σ, γ = log 2, while Lemma 3v entails σ2, γ = / γ see Appendix A.2. Contrary to our previous analyses, it is more difficult to relate these constants to pre-existing results in high quantile or high CTE estimation because Theorem 2 is a general result that applies to a wide range of estimators γ n. To the best of our nowledge, there is no general analogue of this result in the literature for the case p =. In the case p = 2, we find bac the asymptotic distribution result in Theorem of [9]: m2 α n n αn CTEα n d γ ξ λ + ξ 2 γ ρ. It should be highlighted that, for p = 2, Theorem 2 in the present paper is a stronger result than Theorem of [9], since the condition on γ is less stringent than that of Theorem therein. As the above convergence is valid for any γ <, one may therefore thin that the estimator m 2 α n is a widely applicable estimator of the CTE at high levels. Since it is also more robust than the direct empirical CTE estimator due to its reliance on the sample quantile X nαn,n, this would defeat the point of looing for a middle ground solution between the sensitivity of CTE to high values and the robustness of VaR-type measures in very heavy-tailed models. The simulation study in [9] shows however that in general, the estimator m 2 fares worse than the direct CTE estimator m 2, and increasingly so as γ increases within the range 0, /4]. We will also confirm this in our simulation study by considering several cases with higher values of γ and showing that the estimator m 2 should in general not be preferred to m 2. The benefit of using the indirect estimator m p will rather be found for values of p away from 2, when a genuine compromise between sensitivity and robustness is sought. Theorem 2 applies whenever γ n is a consistent estimator of γ that satisfies a joint convergence condition together with the intermediate order statistic X nαn,n. This is not a restrictive requirement. For instance, if γ n = γ H n is the widely used Hill estimator [29]: γ H n := n α n log X n i+,n log X n n αn,n, 9 n α n i= then, under the bias condition n α n A α n λ, we have the joint convergence n αn γ n H γ, X nα n,n d γn + λ qα n ρ, γn 2 where N, N 2 is a pair of independent standard Gaussian random variables for a proof, combine 20

22 Theorem 2.4., Lemma and Theorem in [28]. As a corollary of Theorem 2, we then get the following asymptotic result on m p α n when the estimator γ n is the Hill estimator γ H n. Corollary. Suppose that p and C 2 γ, ρ, A holds with γ < /p. Assume further that α n is such that n α n and n α n A α n λ R. Then, as n : mp α n n αn m p α n d σp, γ N λ Rp, γ, ρ, vp, γ, ρ where vp, γ = γ 2 [σp, γ] 2 +. The asymptotic variance vp, γ is plotted, for several values of p, on Figure 4, and a comparison of the asymptotic variance V p, γ of the direct estimator with vp, γ is depicted on Figure 5, for γ [/4, /2]. Figure 4: Behaviour of the asymptotic variance v of the indirect tail L p median estimator: plot of the function γ vp, γ, on the interval 0, /2, for p {,.2,.4,.6,.8, 2}. It can be seen from these figures that the indirect estimator has a lower variance than the direct one. The difference between the two variances becomes sizeable when the quantity 2γp gets closer to, as should be expected since the asymptotic variance of the direct estimator asymptotically increases to infinity see Theorem, while the asymptotic variance of the indirect estimator is ept under control see Corollary. This seems to indicate that the indirect estimator 2

23 Extreme tail Lp median estimation L. Gardes et al gamma p gamma p Figure 5: A comparison between the asymptotic variances V and v with plots of the function p, γ 7 logv p, γ/vp, γ. Top: on the domain p, γ [,.8] [0.25, 0.5], bottom: on the domain p, γ [.8,.95] [0.25, 0.5]. 22

24 should be preferred to the direct estimator in terms of variability. However, the indirect estimator is asymptotically biased due to the use of the approximation m p α/qα /κp, γ in its construction, while the direct estimator is not. We will see that this can mae one prefer the direct estimator in terms of mean squared error on finite samples, even for large values of γ and p. We conclude this section by mentioning that similar joint convergence results on the pair γ n, X nαn,n, and therefore analogues of Corollary, can be found for a wide range of other estimators of γ. We mention for instance the maximum lielihood estimator in an approximate Generalised Pareto model for exceedances and probability-weighted moment estimators; we refer to e.g. Sections 3 and 4 of [28] for an asymptotic analysis of these alternatives. This can be used to construct several other indirect tail L p median estimators. 4.3 Extreme case: an extrapolation device Both the direct and indirect estimators constructed so far are only consistent for intermediate sequences α n such that n α n. Our purpose is now to extrapolate these intermediate tail L p median estimators to proper extreme levels β n with n β n c < as n. The extrapolation argument is based on the fact that under the regular variation condition C γ, the function t Ut = q t is regularly varying with index γ. In particular, we have: qβ n qα n = U β n U α n βn α n when α n, β n. This approximation is at the heart of the construction of the classical Weissman extreme quantile estimator q W β n, introduced in [47]: γn q W βn β n := X nαn,n. α n The ey point is then that, when γ < /p, the quantity m p α is asymptotically proportional to qα, by Proposition 2. Combining this with the above approximation on ratios of high quantiles suggests the following extrapolation formula: γ βn m p β n m p α n. α n An estimator of the extreme tail L p median m p β n is obtained from this approximation by plugging in a consistent estimator γ n of γ and a consistent estimator of m p α n. In our context, γ 23

25 the latter can be the direct, empirical L p estimator m p α n, or the indirect, intermediate quantilebased estimator m p α n, yielding the extrapolated estimators γn γn m W βn p β n := m p α n and m W βn p β n := m p α n. α n α n We note, moreover, that the latter estimator is precisely the estimator deduced by plugging the Weissman extreme quantile estimator q W β n in the asymptotic proportionality relationship m p β n /qβ n /κp, γ, since γn m W βn p β n = α n { } X nαn,n = qw β n κp, γ n κp, γ n. The asymptotic behaviour of the two extrapolated estimators m W p β n or m W p β n is analysed in our next main result. Theorem 3. Suppose that p and C 2 γ, ρ, A holds with ρ < 0. Assume also that α n, β n are such that n α n and n β n c <, with n α n / log[ α n ]/[ β n ]. Assume finally that n α n γ n γ d ξ, where ξ is a nondegenerate limiting random variable. i If γ < /[2p ] and n α n A α n = O then, as n : n αn m W p β n log[ α n / β n ] m p β n d ξ. ii If γ < /p and n α n A α n λ R then, as n : n αn m W p β n log[ α n / β n ] m p β n d ξ. This result shows that both of the estimators m W p β n and m W p β n have their asymptotic properties governed by those of the tail index estimator γ n. This is not an unusual phenomenon for extrapolated estimators: actually, the very fact that these two estimators are built on an intermediate tail L p median estimator and a tail index estimator γ n sharing the same rate of convergence guarantees that the asymptotic behaviour of γ n will dominate. A brief, theoretical justification for this is that while the intermediate tail L p median estimator is n α n relatively consistent, the estimated extrapolation factor [ β n ]/[ α n ] γn, whose asymptotic behaviour only depends on that of γ n, converges relatively to [ β n ]/[ α n ] γ with the slower rate of convergence n α n / log[ α n / β n ]. This is explained in detail in the proof of Theorem 3, and we also refer to Theorem of [28] and its proof for a detailed exposition regarding 24

26 the Weissman quantile estimator. In particular, if γ n is the Hill estimator 9, then the common asymptotic distribution of our extrapolated estimators will be Gaussian with mean λ/ ρ and variance γ 2, provided n α n A α n λ R, see Theorem in [28]. Let us highlight though that while the asymptotic behaviour of γ n is crucial, we should anticipate that in finite-sample situations, an accurate estimation of the intermediate tail L p median m p α n is also important. A mathematical reason for this is that in the typical situation when β n = /n considered recently by e.g. [3,, 2, 27], the logarithmic term log[ α n / β n ] has order at most logn, and thus for a moderately high sample size n, the quantity n α n / log[ α n / β n ] representing the rate of convergence of the extrapolation factor may only be slightly lower than the quantity n α n representing the rate of convergence of the estimator at the intermediate step. Hence the idea that, while for n very large the difference in finite-sample behaviour between any two estimators of the tail L p median at the basic intermediate level α n will be eventually wiped out by the performance of the estimator γ n, there may still be a significant impact of the quality of the intermediate tail L p median estimator used on the overall accuracy of the extrapolated estimator when n is moderately large. This is illustrated in the simulation study below. 5 Simulation study Our goal in the present section is to assess the finite-sample performance of our direct and indirect estimators of an extreme tail L p median, for p [, 2]. In addition, we shall do so in a way that provides guidance as to how an extreme tail L p median, and its estimates, can be used and interpreted in practical setups. Let us recall that our focus is not to consider cases with low γ, as in such cases the easily interpretable CTE ris measure can be used and estimated with good accuracy, including at extreme levels. We shall rather consider cases with γ > /4, where the fact that the tail extreme tail L p median realises a compromise between the robust MS and the sensitive CTE should be expected to result in estimators with an improved finite-sample performance compared to that of the classical empirical CTE estimator. It was actually highlighted in 3 and 4 that, for p [, 2] and γ < /p, an extreme tail L p median m p α can be understood asymptotically as a weighted average of MSα and CTEα. In other words, defining 25

27 the interpolating ris measure R λ α := λmsα + λcteα, we have m p α R λ α as α with λ = λp, γ as in 4. It then turns out that at the population level, we have two distinct but asymptotically equivalent possibilities to interpolate, and thus create a compromise, between extreme Median Shortfall and extreme Conditional Tail Expectation: Consider the family of measures m p α, p [, 2]; Consider the family of measures R λ α, λ [0, ]. In the rest of this section, based on a sample of data X,..., X n of size n, we consider the estimation of the tail L p median m p α n for both intermediate and extreme levels α n, and how this estimation compares with direct estimation of the interpolating measure R λ α n. 5. Intermediate case We first investigate the estimation of m p α n, for p [, 2], and of R λ α n, for λ [0, ], in the intermediate case when α n and n α n. As far as the estimation of m p α n is concerned, we compare the direct estimator m p α n defined in 6 and the indirect estimator in 8. In the latter, γ n is taen as the Hill estimator defined in 9: in other words, m p α n = with γ H n = X n n αn,n κp, γ H n n α n logx n i+,n logx n,n. i= To further compare the performance of these two estimators, and therefore the practical applicability of the measure m p α n for interpolating between extreme MS and extreme CTE, the finite sample behaviour of these estimators are compared to that of the estimator of R λ α n given by R λ α n := λx n n αn/2,n + λĉteα n with ĈTEα n = m 2 α n = n α n X n i+,n. n α n i= 26